Author : Noam Greenberg
Release : 2006
Genre : Mathematics
Kind : eBook
Book Rating : 857/5 ( reviews)
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Book Synopsis The Role of True Finiteness in the Admissible Recursively Enumerable Degrees by : Noam Greenberg
Download or read book The Role of True Finiteness in the Admissible Recursively Enumerable Degrees written by Noam Greenberg. This book was released on 2006. Available in PDF, EPUB and Kindle. Book excerpt: When attempting to generalize recursion theory to admissible ordinals, it may seem as if all classical priority constructions can be lifted to any admissible ordinal satisfying a sufficiently strong fragment of the replacement scheme. We show, however, that this is not always the case. In fact, there are some constructions which make an essential use of the notion of finiteness which cannot be replaced by the generalized notion of $\alpha$-finiteness. As examples we discuss bothcodings of models of arithmetic into the recursively enumerable degrees, and non-distributive lattice embeddings into these degrees. We show that if an admissible ordinal $\alpha$ is effectively close to $\omega$ (where this closeness can be measured by size or by cofinality) then such constructions maybe performed in the $\alpha$-r.e. degrees, but otherwise they fail. The results of these constructions can be expressed in the first-order language of partially ordered sets, and so these results also show that there are natu
